216 XIII INTEGRATION for the injection JS?c (X, ju) -» 3? c(X, ju) (with respect to the corresponding seminorms) because it follows from (13.12.5) that N2(/) ^ N^/) • MX)1/2 for all/e.Sfj?(X,/0. (13.20.5) In particular (13.9.17), if /j is a bounded (complex) measure, every function fe%g(X) is 0-integrable, and we have I jfdfj, g HI • ||/|| by (13.16.5). In other words, / i-> |/ d^ is a continuous linear form on the Banach space ^c W- But it should be noted that, in general, there exist continuous linear forms on this space which are not of this type. (1 3.20.6) The space M£(X) is also the dual of the closure ^g(X) of Jf C(X) in the Banach space #£(X) (12.15). A function / belonging to *g(X) may be characterized by the following property :for each s > 0, there exists a compact subset K ofX such that |/(jc)| ^ sfor allxeX-K. For if/e «?(X), then for each & > 0 there exists by definition a function g e ^TC(X) such that ||/— #|| :gj e; if K is the support of g, then \f(x)\ ^ e for all x $ K. Conversely, suppose that / has the above property, and let h be a continuous mapping of Xinto [0, 1], with compact support and equal to 1 on K ((3.18.2) and (4.5.2)); it is clear that \\f-fh\\ £ s and that//ze Jfc(X), hence /e «g(X). The functions belonging to ^cPO are called (complex-valued) continuous functions which tend to 0 at infinity. (When X = R, they are indeed the continuous functions / such that lim /(#) = lim f(x) = 0.) We put n ^R(X) for the corresponding space of real-valued functions. When X is compact, we have PROBLEMS 1. The space Mi(X) of bounded real measures on a locally compact space X can be considered as a space of linear forms on each of the following vector spaces : (1) the space EI = ^TR(X) of continuous functions with compact support; (2) the space E2 = #a(X) of continuous functions which tend to 0 at infinity; (3) the space E3 = ^f(X) of bounded continuous functions; (4) the space E5 of linear combinations of (upper or lower) semicontinuous bounded functions; (5) the space E6 = ^n(X) of universally measurable bounded functions. Moreover, if v is any positive measure on X, the space Mi(X, v) of bounded measures with base v (which may be identified with the space L£(X, v) by virtue of 13.14.4)) can be considered as a space of linear forms on the vector space E4,v of bounded functions which are continuous almost everywhere with respect to v. y fn is everywhere finite and continuous (because every point